GMAT Question 152

ironsheep

Junior Member
Joined
May 9, 2019
Messages
106
This is a question from "GMAT Official Guide 2019 Quantitative Review" and it says, "A certain university will select 1 of 7 candidates eligible to fill a position in the mathematics department and 2 of 10 candidates eligible to fill 2 identical positions in the computer science department. If none of the candidates is eligible for a position in both departments, how many different sets of 3 candidates are there to fill the 3 positions?" The answer is 315

1. I did 7 times 10 times 9 equals 630. 630/3 equals 210. For the first job, any 7 will do. For the second any 10 will do and for the third any 9 will do, so multiply them all and you 630. Since they want sets of three, you do 630/3 = 210.

2. 7 plus (10 times 9) == 97. You don't mulitply 7, 10, and 9 because the math department is different from the computer science department. You can just add them and I didn't bother to divide by 3 because the correct answer is 315 not 97.

I don't understand what they mean when they wrote, "If none of the candidates is eligible for a position in both departments, how many different sets of 3 candidates are there to fill the 3 positions?"
 
The questions that you have shown have rather consistently been badly written. I tutor English as well as math so I know something about clear writing. So I agree with you that the question is not written as clearly as it could be, but a lot of writing is pretty godawful so you just have to do your best to figure out what it means. I have become rather negative about the competence of the GMAT folks, but if that is the test you must take, you will have to cope.

Here is what I think they mean.

There are seven individuals eligible to fill the single position in the mathematics department. There are ten individuals eligible to fill two positions in the computer science department. The individuals eligible to work in the mathematics department are not eligible to work in the computer science department, and the individuals eligible to work in the computer science department are not eligible to work in the mathematics department. How many sets of three individuals are there to consider for filling the three positions?

Now, let's consider the answer.

How many singletons can you form from a set of seven?

[MATH]\dbinom{7}{1} = \dfrac{7!}{1! * (7 - 1)!} = \dfrac{7 * 6!}{1! * 6!} = 7.[/MATH]
You got that correct, which is simple common sense. But your 90 was wrong. You were in essence counting Tom and Mary as a different pair than Mary and Tom.

How many doubletons can you form from a set of ten?

[MATH]\dbinom{10}{2} = \dfrac{10!}{2! * (10 - 2)!} = \dfrac{10 * 9 * 8!}{2 * 8!!} = \dfrac{90}{2} = 45.[/MATH]
[MATH]7 * 45 = 315.[/MATH]
 
You got that correct, which is simple common sense. But your 90 was wrong. You were in essence counting Tom and Mary as a different pair than Mary and Tom.


Can you please explain this more, please? How am I double counting each person for the computer department.
 
You were in essence counting Tom and Mary as a different pair than Mary and Tom.

Okay, I got that part, it just took a while to process it. When I see "how many different sets of 3 candidates are there to fill the 3 positions" I think of dividing by three somewhere, why is this thinking wrong?
 
Okay, I got that part, it just took a while to process it. When I see "how many different sets of 3 candidates are there to fill the 3 positions" I think of dividing by three somewhere, why is this thinking wrong?

Because this is an example of intuition, not thinking. There must be a reason to conclude that you have to divide a particular value by 3.
 
I am not sure that I fully agree with lev although he is without a doubt better at mathematics than I am.

The problem with making arguments from intuition is that intuition changes with experience and thought. Einstein had better intuition about physics than I ever will. So no one should give a rat's ass about my intuition about physics. (On the other hand, I suspect my intuition about the psychology of negotiating is a whole lot better than Einstein's would have been.)

I shall try to make an informal argument, but ultimately the only arguments that can be relied on are those based on logic and mathematics.

Let's say Alex is a member of the mathematical pool. Let's say Tom and Mary are members of the computer science pool. If we care about the order in which the members were selected, then

AMT
ATM
MAT
MTA
TAM
TMA

are all different, and choosing just those 3 people represents 6 different sequences. But we do not care about the order in which they were selected, we care only about those who were selected. So if we are going to divide by anything, it is 6, not 3.

So let's think along the lines of order of selection and divide by 6.

We can choose mathematician, computer scientist, computer scientist.

[MATH]7 * 10 * 9 = 630.[/MATH]
We can choose computer scientist, mathematician, and computer scientist.

[MATH]10 * 7 * 9 = 630.[/MATH]
We can choose computer scientist, computer scientist, and mathematician.

[MATH]10 * 9 * 7 = 630.[/MATH]
[MATH]\dfrac{630 + 630 + 630}{6} = \dfrac{3 * 630}{3 * 2} = \dfrac{630}{2} = 315.[/MATH]
Now once you have done enough problems where order is irrelevant, you learn how to use binomial coefficients to save a whole bunch of time.

[MATH]\dbinom{7}{1} * \dbinom{10}{2} = \dfrac{7!}{1! * 6!} * \dfrac{10!}{2! * 8!} = \dfrac{7 * 10 * 9}{1 * 2} = 315.[/MATH]
Intuition depends on knowledge and familiarity. I suspect I shall manage my second marriage better than my first.

EDIT: My wife upon reading this says that I had two advantages in my first marriage that will not obtain in my next: (a) I was a LOT cuter in my twenties than I am in my seventies, and (b) in my next marriage, I shall not have the advantage of being married to her.
 
Last edited:
although he is without a doubt better at mathematics than I am.
Now, this is an example of a completely baseless statement.

but ultimately the only arguments that can be relied on are those based on logic and mathematics.
That's what I was trying to say, so not sure where we disagree.
 
Now, this is an example of a completely baseless statement.
Nah, I do not consider that I am a mathematician at all. My rank in mathematics is zero; my academic training is in European history and languages.

That's what I was trying to say, so not sure where we disagree.
Where we disagree is that intuition is a relative standard. Not a big disagreement because we ultimately agree that intuition is not reliable. I just shy away from asking students to rely AT ALL on an intuition that is is necessarily very limited.
 
I am not sure that I fully agree with lev although he is without a doubt better at mathematics than I am.

The problem with making arguments from intuition is that intuition changes with experience and thought. Einstein had better intuition about physics than I ever will. So no one should give a rat's ass about my intuition about physics. (On the other hand, I suspect my intuition about the psychology of negotiating is a whole lot better than Einstein's would have been.)

I shall try to make an informal argument, but ultimately the only arguments that can be relied on are those based on logic and mathematics.

Let's say Alex is a member of the mathematical pool. Let's say Tom and Mary are members of the computer science pool. If we care about the order in which the members were selected, then

AMT
ATM
MAT
MTA
TAM
TMA

are all different, and choosing just those 3 people represents 6 different sequences. But we do not care about the order in which they were selected, we care only about those who were selected. So if we are going to divide by anything, it is 6, not 3.

So let's think along the lines of order of selection and divide by 6.

We can choose mathematician, computer scientist, computer scientist.

[MATH]7 * 10 * 9 = 630.[/MATH]
We can choose computer scientist, mathematician, and computer scientist.

[MATH]10 * 7 * 9 = 630.[/MATH]
We can choose computer scientist, computer scientist, and mathematician.

[MATH]10 * 9 * 7 = 630.[/MATH]
[MATH]\dfrac{630 + 630 + 630}{6} = \dfrac{3 * 630}{3 * 2} = \dfrac{630}{2} = 315.[/MATH]
Now once you have done enough problems where order is irrelevant, you learn how to use binomial coefficients to save a whole bunch of time.

[MATH]\dbinom{7}{1} * \dbinom{10}{2} = \dfrac{7!}{1! * 6!} * \dfrac{10!}{2! * 8!} = \dfrac{7 * 10 * 9}{1 * 2} = 315.[/MATH]
Intuition depends on knowledge and familiarity. I suspect I shall manage my second marriage better than my first.

EDIT: My wife upon reading this says that I had two advantages in my first marriage that will not obtain in my next: (a) I was a LOT cuter in my twenties than I am in my seventies, and (b) in my next marriage, I shall not have the advantage of being married to her.


So the reason why it is (90/2) is because order of candidates don't matter. Okay and since I got the first part right with the number being 7, then it is just 7 times 45, which is 315.
 
So the reason why it is (90/2) is because order of candidates don't matter. Okay and since I got the first part right with the number being 7, then it is just 7 times 45, which is 315.
Yes.

Given the questions you will be facing, you need to review counting principles, particularly permutations and combinations. It's not hard. An hour or two of thinking, and a few hours of doing problems.
 
Nah, I do not consider that I am a mathematician at all. My rank in mathematics is zero; my academic training is in European history and languages.
I completed 2.5 years of applied math 30 years ago, but since then I've been programming, so most of what I learned I haven't practiced. We all know what happens in such cases :)

Where we disagree is that intuition is a relative standard. Not a big disagreement because we ultimately agree that intuition is not reliable. I just shy away from asking students to rely AT ALL on an intuition that is is necessarily very limited.
I'll try to clarify. I pointed out that "I think of dividing by three somewhere" is intuition, not thinking and, therefore, should not be used in solving math problems. Are you saying that it _can_ be used?
 
I completed 2.5 years of applied math 30 years ago, but since then I've been programming, so most of what I learned I haven't practiced. We all know what happens in such cases :)


I'll try to clarify. I pointed out that "I think of dividing by three somewhere" is intuition, not thinking and, therefore, should not be used in solving math problems. Are you saying that it _can_ be used?
Ahh sorry. I was not trying to pick a fight. I misread the intent of your first comment. I read it as saying that the way to attack the problem was through intuition rather than the application of counting principles. I apologize.
 
Top